Decimal multiple and sub-multiple units of physical quantities

According to the information given by engineering psychology the human being perceives the numbers from 0,1 to 1000 best of all. The PQ magnitudes range, used in the scientific and production practice, greatly exceeds this interval.

That's why for the facilitation of the PQ magnitudes perception the decimal multiple and sub-multiple units of PQ have been introduced. The multiple units are obtained by multiplication of the initial units by 10, raised to the positive power, and sub-multiple - to the negative one (Table 3).

 The prefix or its designation should be written indissoluble with the name of the initial unit, or its designation, respectively. When the unit is formed as the multiplication or ratio of units, it is necessary to join the prefix to the name of the first unit. For example, the right variant is: kilopascal - second divided by meter (kPa · s/m); the wrong variant is: - pascal - kilosecond divided by meter (Pa · ks/m). It is allowed to add the prefix to the second multiplier or denominator only in cogent cases, when these units are widely spread and the substitution by the other ones is complicated and unusual. For example: tonne-kilometre (t · km); ampere divided by square millimeter (A/mm²).

It is not allowed to attach two or more prefixes to the name of the unit. For instance in the past the unit of electric capacitance micromicrofarad had been in use. In accordance with the modern rules it must be called now "picofarad". To minimize mistakes during calculations in multiple and sub-multiple units it is recommended to enter only the final result, and in the act of calculation all quantities must be presented in SI units, with the prefixes replaced by multipliers by 10 in appropriate powers.

Table 3

Formation of decimal multiple and sub-multiple units

Relative and logarithmic units

Relative unitis a ratio of two units the same name, one of which is accepted as an initial.

Percent, %, is one hundredth part of the whole. Per mil(le),%, is one thousandth part of the whole.

Logarithmic unit isa logarithm (decimal, natural or with base 2) of a dimensionless ratio of a PQ to the PQ of the same name, which magnitude is accepted as initial. For the PQs by dimension like energy, power, work, etc., the following formula is used:

         under ;

under     

 where and are the quantities of the same name, for example, power, energy, work, etc.

 For the PQs by dimension like value of electric current, voltage, pressure, etc:

 1Bel = 2 lg(F₂/F₁)  under ;

under   

where and are the PQs of the same name of the above mentioned nature.

In practice decibel, the tenth part of bell,is used mostly; the international designation is dВ, and Ukrainian is дБ.

For example, voltage gain, ratio of signal to noise are measured in decibel, etc.

 It is useful to memorize, that for the voltage, current, force, etc. gain of 1 dВ is equal approximately to 10 % , 6 dВ is equal to 2 times, 10 dВ is equal to 3,16 times, 20 dВ is equal to 10 times, 30 dВ is equal to 31,6 times, 40 dВ is equal to 100 times, etc.

For example, for a three-stage voltage amplifier with gains of the cascades respectively 10 dВ, 26 dВ and 30 dВ the total gain is:

10 dВ + 26 dВ + 30 dВ = 66 dВ,

i.e., 2000 times (60 dВ is equal to the gain in 1000 times , and 6dВ - in two times ).

The voltage is often measured in decibel, and the magnitude of 0,775 V is taken as the initial voltage level, i.e. as zero decibel.

For example, the voltage level is 40 dВ. How to convert it to the volts?

 where:

Ux - is the unknown voltage; U₀ - is the initial level (0,775 V).

Hence: , i.e. ,

And:   V.

When measuring the sound intensity (loudness) by the logarithmic units scale, the sound oscillations pressure of 20 mPa is taken as the initial level, i.e. as zero decibel (this is near the human acoustic perception threshold).

The loudest sounds, which a human can perceive without damage of ability to hear, reach 120 dВ by this scale (pressure 20 Pa). 

Sounds of this kind are created by engine of a jet plane at a distance of several metres from it, a thunderpeal just over head, etc.

Pay your attention to the conveniance of the logarithmic scale. 

Instead of applying the scale of inconceivable range from 20 mPa to 20000000 m Pa we have a handy scale from 0 to 120 dВ.

When measuring frequency bands the following logarithmic units as octave and decade [3; 4] are used:

where and are the lower and the upper frequencies of the range under research , respectively.

Questions for self-verification

1. What is metrology? What major theoretical and practical problems does it deal with?

2. What is PQ? What is the difference between the three levels of its value used in the metrological terminology?

3. What base and supplementary PQ units form the SI? How are they defined?

4. What are the most important derived PQ units in the field of electricity and magnetism, that form the SI? How are they defined?

5. What are the prefixes of the decimal multiple and submultiple PQ units? What are the rules of their usage?

6. For which measurements are the relative and logarithmic units used? What are their advantages? According to what rules are they used?

Forms of measurements

Measurement is representation of the physical quantities being measured with their values by means of experiment and calculations by means of special technical facilities.

It is impossible to obtain a measurement result only by means of some mathematical operations with some physical constants without physical experiment.

It is also impossible to regard as a measurement result, for example, an allegation, that a distance between the objects is equal to 10 steps, since the special technical instruments, for example, a tape-measure, were not used.

And the step size of various people is not identical and standardized.

As a final measurement result can be reached by different methods, the measurements can be divided into the following forms.

Direct measurement is a measurement of one quantity, when its value is determined directly, without transformation of its kind and use of known correlations.

 For example, measurement of voltage by voltmeter, resistance by ohmmeter, power-factor by phasemeter, etc.

Indirect measurement is a measurement, when value of one or several quantities under measurement are determined after transformation of the quantity kind or calculations by known correlations. The indirect measurement, in turn, can be divided into three types: simple indirect (or inferential) measurement, measurement in a closed series (or cumulative measurement) and a simultaneous measurement.

The indirect (or inferential) measurement is an indirect measurement of one quantity with transformation of its kind or calculation of other quantities measurement results, with which the measurand is correlated by an explicit functional relation .

For example, if necessary to measure an area of a rectangular table, one should determine its length and width by means of the direct measurements. Then, having used a rectangle area dependence on its length and width, known from the course of mathematics, one can calculate the table area.

 Resistance of a resistor can also be determined by direct measurement of the current flowing through it with an ammeter and a voltage drop across it with a voltmeter. These three physical quantities are interconnected through the well-known Ohm’s law.

The measurement in a closed series (or cumulative measurement), is an indirect measurement, when the values of several congeneric quantities being measured simultaneously are obtained by solution of the equations, that connect the different combinations of these quantities, which are measured directly or indirectly.

For example, we have a kit of ten weights, by mass from 1 to 10 g; only one weight has a mass mark. To determine a mass of each of other nine weights, let us compare their different combinations by means of a balance.

Let us assume, that the comparisons yield the following equations: 

                                                                etc.

where m_i  is mass of the corresponding weight from kit.

Having obtained not less than 9 equations, and having solved their system, we can find the mass of each weight from the kit .

      One more example of an electrical quantity measurement (Fig.2).

Inside a closed box three resistors are jointed by “star”, their unknown resistances are designated as R₁₀, R₂₀ and R₃₀. Three leads are

brought out (points 1, 2, 3). There is no access into the box to the resistor’s junction point. We have an ohmmeter for measurements.

As it is impossible to determine the resistances R₁₀, R₂₀ and R₃₀ by means of the direct measurements, the only way is to apply the measurement in a closed series.

Fig.2. The measurement in a closed series of the resistor resistances

For this purpose the resistances between points 1; 2 and 3, i.e. R₁₂, R₂₃ and R31 are measured by ohmmeter; having used the scheme (fig.2), we write a set of equations:

Having solved this system, we get the required resistances R₁₀, R₂₀ and R₃₀

The simultaneous measurement is an indirect measurement, when the values of several heterogeneous quantities measured simultaneously are obtained by solution of equations, connecting them with other quantities, which are measured directly or indirectly.   

For example, it's known, that a resistor’s resistance depends on temperature:

                     ,

where R_ti is a resistor’s resistance at i-th temperature; R_to is a

 resistor’s resistance at some initial, for example, zero, temperature; α is a temperature coefficient:

                    ,

where t_i is a temperature, at which we can directly measure a resistor’s resistance; t₀is initial temperature.

We can measure the temperature t_i directly by a thermometer, and the resistance R_ti - by an ohmmeter. What is wanted is R_to and α.

We measure directly the resistor’s resistance at two different temperatures at least:

Having solved this set, we obtain the unknown R_to and α, thus we can calculate the resistance of this resistor at any temperature.

Let us consider one more example, which shows, that in some cases, we can get the required quantities only by means of the simultaneous measurement.

It is known, that magnetic core losses P are the sum of the losses due to hysteresis, Р_g, and the losses due to eddy-currents, Р_e: 

P = P_g + P_e ,

moreover, only the total losses, P, can be measured directly, e.g. by a wattmeter.

To determine the Р_g and P_e separately, let us make use of their different frequency dependences:                                 

                       P_g = af;   P_e = bf²,

where f is a current frequency ; aand bare coefficients, constant for the given magnetic core.

These coefficients can be determined just by measuring the total losses at various frequencies.

Heaving solved the system, we can get the coefficients a and b for the given magneti core. As a result, we can calculate the hysteresis losses and the eddy-currents losses separately at known current frequency ¦.

Methods of measurements

Each form of measurements mentioned above can be realized by different methods.

Method of measurementis a set of application methods of measurment instruments and principles to derive a measurement information.

Principle ofmeasurementis a complex of physical phenomenas, on which the measurement is based.

All methods of measurements, depending on the use of a material measure during a measurement experiment, can be divided into two groups:

1) methods of measurements by comparison against an actual measure;

2) direct evaluation method .

In the first group, a material measure (a device that reproduces a PQ of fixed value) participates in each measurement experiment. For example, a weighing on a balance with equilibration of unknown mass by the weights (by the mass material measures).If we use the method of direct evaluation, a material measure is absent during the measurement experiment. For example, weighing with dial-indicating scales, when the unknown mass is determined by a pointer position over a calibrated scale, i.e., without the use of the weights.

It may appear, that it is possible to measure without material measure at all. But this is not true.The weights (the mass material measures) were used to calibrate the dial-indicating scales; their action has been kept in mind by a spring and a scale. Later on, during any

 following weighing this "memory" reproduces the action of the weights.

That’s why, in the old textbooks the terms “simultaneous” and “occurring in different times” comparison against a material measure were used for these groups of measurement methods, respectively.

Nowadays the following definitions are accepted:

The method of measurement by comparison against an actual measureis a method of measurement, in which a measurand is compared with a quantity, that is reproduced by a material measure.

The method of direct evaluation is a method of measurement, in which a value of physical quantity is determined directly from an indicating device of a direct-conversion instrument.

The direct evaluation method is the most widespread in practice. It is simple, convenient, takes a little time, does not require high operator’s qualification. But the accuracy of this method is limited.

The method of measurement by comparison against an actual measure has peculiar properties, opposite to the above-enumerated, the most important of them is higher accuracy.

It’s also in common use in the following most generally employed varieties.

The differential method of measurementis a method of measurement, in which a small difference between a measurand and on output magnitude of a single-channel material measure is measured by a suitable measurement instrument.

 For example, we have dial-indicating scales with operating range of 500 g (Fig.3), and it is necessary to weigh a mass about 600 g. For this purpose we put the 500 g weights, on the left scale of the scales, and the unknown mass on the right scale. If the pointer indicates of 100 g, it means that the object weighed has mass of 600 g. Here the dial-indicating scales measures only a small difference between mass of 600g, which is under measurement, and the weight mass of 500 g.

This is more accurate, than the measurement of the 600 g mass, for example, by scales of 1000 g operating range. The fact is, that with the differential method the dial-indicating scales (it is, as a rule, a measurement instrument of not high accuracy) weigh not the whole mass of 600 g, but only its small part, therefore, we have an error practically only of this small part. The error of the part compensated by the weight (500 g) is incomparably small due to the high accuracy of the weight.

Fig.8. The differential method of measurement:

m_x is an unknown mass; m₀ is a mass of a material measure (a weight); Δ is an indication of the dial- scales.

When weighing the mass of 600g by the direct evaluation method with dial- scales of 1000g operating range, the scales measures the whole mass of 600g, and an error from the whole mass of 600g, but not from the small difference of 100g becomes the measurement result error.And this is well over, than in the previous case.

The null method of measurementsis a method of direct measurement with reiterated comparison of a measurand against a quantity that is reproduced by a material measure, which is regulated to their complete equilibration. For example, if in the illustration of the differential method application for weighing mass of 600g considered before, we put the 600g weights on the opposite scale of scales, so that the pointer over the scale indicates 0, this is the null-method.

Typical examples of the null method application are weighing by a scale- less balance (Fig.4), measurement of electric resistor resistance by means of a bridge, etc.

The null-method of measurement is the most accurate of all known methods.

 Fig.4. Null-method of measurement; mx is unknown mass; moi is the total mass of the weights (mass material measures), that balances the unknown mass

The substitution method of measurement is a method of indirect measurement with frequentative comparison to a complete equlibration of the output quantities of a measurement transducer that transforms both a measurand and the output quantity of a regulable material measure by turns. This is one of the comparison methods, in which a measurand is replaced by the known quantity that is reproduced by a material measure.For example, it is necessary to measure an electric resistor resistance, and we have only one measuring instrument – an ammeter.

For this purpose let us compose a circuit with a series connection of this resistor and an ammeter to a supply source (Fig.5).

Fig.5. The substitution method: Е is a voltage source; Rx is a resistor of unknown resistance; Roi is a known, regulable resistance of a resistance box (resistance material measure)

Having taken down the ammeter indication, we replace this resistor with a resistance box. By varying resistance of the resistance box, we attain the previous ammeter indication. Having read the resistance of the box, we can determine the resistance of the resistor examined.

The coincidence method of measurement is a comparison method, in which difference between a measurand and a quantity that is reproduced by a material measure is measured by using coincidence of the scale marks or the periodic signals.

For example, it is necessary to determine size of inch on an inch-marked scale rule with a millimeter marked rule. For this purpose it is necessary to adjoin the rules one to another in such a way, that their zero marks have coinsided. All other marks do not exactly coincide, except for some, namely 5 inches and 127 mm, 10 inches and 254 mm, etc. By using these coincidences one can determine the necessary size.

All nonius scales of caliperses and micrometers are based on the coincidence method (Fig.6). Measurement of rotation frequency by a stroboscope is found by the same method. In this case we can observe mark places coincidence on a rotating object, at the flash moments of the known controlled frequency.

Fig.6. The coincidence method. The calipers nonius scale (Dx=22,5mm)